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The Polynomial van der Waerden Theorem

[Bill Gasarch, known online for his frequent contributions to the Complexity Blog and for his liberal USE of capital LETTERS, has been reading about the “polynomial” and “multi-dimensional” versions of van der Waerden’s theorem (the original theorem being “linear” and “one-dimensional”) and he is going to tell us about it. Enjoy. — Luca]

Guest Posting by William Gasarch (gasarch@cs.umd.edu)

This is a one of two Guest Posting on the Polynomial van der Waerden Theorem (henceforth Poly VDW). This posting is on the one-dimensional Poly VDW; however,my next posting will be on the Multidimensional Poly VDW. Throughout this posting `Poly VDW’ means `One Dimensional Poly VDW’

Our starting point is VDW’s theorem, which is the same starting point Luca had in his postings on Szemeredi’s theorem. (here, here, here, here, and here.) But I go in a different direction.

VDW’s:For all , for any -coloring , there exists , , such that

The original proof ofVDW’s Theorem was an induction on the following ordering on

For example the proof of depends on the theorem being true for where is very large. Shelah (Primitive recursive bounds for van der Waerden numbers, Journal of the American Math Society, Vol 1, 1-15, 1988) has a proof that avoids this type of induction and hence yields a much slower growth rate for the VDW numbers (which we are not discussing here).

Consider the functions

One may consider replacing with polynomials in . This leads to the following theorem.

Poly VDW:For all , for all such that , for any -coloring there exists , , such that

The condition is needed since if the ‘s were constants the theorem would be false. It is an open problem to look at particular sets of polys with constant term.

For every finite set such that we associate a tuple where the largest degree polynomial in is degree , and, for all , , is the number of different coefficients of that occur in polynomials of degree in .

Let mean that Poly VDW holds for all sets of polys of type . The ordinary VDW theorem can be viewed as .

The Poly VDW is proven by induction on the following -ordering. Let .

If then .

If then order lexicographically. (e.g., )

Having stated the Poly VDW one can now state a version of VDW over any infinite Commutative Ring (think Reals).

Gen Poly VDW:Let any infinite commutative ring. For all , for all such that , for any -coloring there exists , , such that

Coda 2:The Maryland Math Olympiad is in two parts. Part I is 25 multiple choice questions (but some are quite hard). If you do well on part I then you can take part II which is 5 problems that need proof. We try to number the problems in order of difficulty. I placed the following problem on the 2006 High School Maryland Math Olympiad, as problem 5:

Let be a 3-coloring of . Show that there exists such that and is a square.

Of the 240 students who took the exam around 180 tried this one. Of those 5 got it right and 10 got partial credit. The proof I had in mind (which is the one the students used who got it right) is not a scaled down version of the proof of the poly VDW. I leave it to the reader to see if they can prove this.

5 thoughts on “The Polynomial van der Waerden Theorem”

AH- the exact wording did not allow x=y.It was“Each positive integer is assigned oneof three colors. Show that there existdistinct integers x,y such that x and yhave the same color and |x-y| is a perfectsquare.”

At the last minute the committee took outthe 2006 stuff- I prefer it with3-coloring {1,…,2006}.

(Bill G here in case it accidentallysays that I am anonymous)YES the theorem is true forany k. That is, if you k-color N YES there will bex,y different |x-y| a square,x,y same color. This is easycorollary of poly vdw theoremin the posting. BUT there is aGOOD question here: is therea HS-level proof of this fork=4? general k? I have not beenable to find one. (Most HS studentdo not know the poly vdw theoremand the proof is not something they could come up with.)SO-if someone can come up witha HS level proof of k=4 case,I would be very interested.